Geodesic
Embedding theorems and best approximations
Sbornik. Mathematics, Tome 26 (1975) no. 2, pp. 213-224

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We establish necessary and sufficient conditions, in terms of best approximations, for a function in $L^p(0,2\pi)$ ($0$) to belong to $L^q(0,2\pi)$ ($q$). The proofs depend on the properties of equimeasurable functions, which were applied by Ul'yanov in the theory of the embedding of certain classes $H_p^\omega$ for $p\geqslant1$ (RZhMat., 1969, 2B109). We also obtain some relationships among moduli of continuity in different metrics, which let us generalize results of Hardy and Littlewood (Math. Z., 28, № 4 (1928), 612–634) to the case $0$ and prove converses for nonincreasing functions. Bibliography: 11 titles. 
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     author = {\`E. A. Storozhenko},
     title = {Embedding theorems and best approximations},
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È. A. Storozhenko. Embedding theorems and best approximations. Sbornik. Mathematics, Tome 26 (1975) no. 2, pp. 213-224. http://geodesic.mathdoc.fr/item/SM_1975_26_2_a3/